What is the general solution for arctan(x+1/3)+arctan(x-1/3)=arctan(2) ?

The general solution for arctan(x+1/3)+arctan(x-1/3)=arctan(2) is x= 2/3

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arctan(x+1/3)+arctan(x-1/3)=arctan(2)

arctan (x + \frac{1}{3}) + arctan (x - \frac{1}{3}) = arctan (2)

x = \frac{2}{3}

Solution steps

arctan (x + \frac{1}{3}) + arctan (x - \frac{1}{3}) = arctan (2)

Rewrite using trig identities

arctan (\frac{x + \frac{1}{3} + x - \frac{1}{3}}{1 - ( x + \frac{1}{3} ) ( x - \frac{1}{3} )}) = arctan (2)

Apply trig inverse properties

\frac{x + \frac{1}{3} + x - \frac{1}{3}}{1 - ( x + \frac{1}{3} ) ( x - \frac{1}{3} )} = 2

Solve

\frac{x + \frac{1}{3} + x - \frac{1}{3}}{1 - ( x + \frac{1}{3} ) ( x - \frac{1}{3} )} = 2 : x = - \frac{5}{3}, x = \frac{2}{3}

x = - \frac{5}{3}, x = \frac{2}{3}

Verify solutions by plugging them into the original equation

x = \frac{2}{3}

sin (θ) = 0, 64

cos (3 x) = cos (2 x)

csc (x^{2}) = \frac{4}{3}

2 cos (x) cos^{3} (x) + cos^{2} (x) - sin^{2} (x) = cos (x)

tan (θ) = \frac{2 3}{- 2}

What is the general solution for arctan(x+1/3)+arctan(x-1/3)=arctan(2) ?
The general solution for arctan(x+1/3)+arctan(x-1/3)=arctan(2) is x= 2/3